The absolute space is a 3-dimensional geometric space U. Every event is mapped to exactly one point in U. In other words, there exist absolute spatial relationships (e.g. distance) among events.
In this section, "location" is meant again in the general sense, i.e. not as point-like location only.
We saw in classical mechanics that there is no known way to permanently mark the locations of absolute space. We can only make sure that the locations are identified momentarily, via the coordinates of any one reference frame. In spite of this shortcoming, the following argument still strongly supports the idea of absolute space in classical mechanics:
Imagine that time was frozen at an (absolute) moment. Then, for that one moment, all observers in every reference frame would see the very same space. All physical objects, as well as the locations they naturally mark, would have the exact same shapes, sizes and arrangement, for each individual observer. This way, in classical mechanics the absolute space can be exhibited.
In special relativity, however, this thought experiment cannot be carried out. Due to time dilation, freezing the "present moment" in one inertial frame would freeze a continuum of moments in any other inertial frame that is moving relative to it. If we decide not to require that the very same moment (i.e. coordinate time) gets frozen at all points of an inertial frame, it would still not be possible, due to length contraction, that the distance between any two given particles is the same in all (frozen) inertial frames. And even if there was a frozen inertial frame whose spatial relationships did in fact match those of the absolute space, there would be no known way to tell which one it is.
At this point, similarly to the case of absolute time, we must realize that we cannot exhibit absolute space in a tangible way, and thus the only thing left in support of it is our outdated intuition. Again, as the principle of relativity suggests that the situation of all inertial frames must be symmetrical, the choice has been made in special relativity to give up absolute space.